First and foremost, it is important to note that all calculations in Dips regarding pole densities, etc, take place ON THE REFERENCE HEMISPHERE, and NOT on the 2-dimensional stereonet. This avoids all of the problems associated with the various methods of determining contours directly on the 2D stereonet, and is in fact much simpler to program and understand.

In Dips, the pole concentration contours are determined as follows:

First, a regular, square grid is superimposed on the 2D stereonet. (In the old DOS version of Dips, the grid spacing could be set to one of two values: 4% or 6% of the projection radius. In the Windows version of Dips, the spacing is fixed at 4%, since a finer grid spacing gives smoother and more accurate contours).

This square grid of points is then back-mapped onto the reference hemisphere, resulting in a corresponding grid of points on the surface of the sphere. This is the actual grid used in the contouring calculations.

Each pole is assigned an influence on the surface of the sphere, according to the Distribution method used - Fisher or Schmidt.

A "floating cone" is then used, such that its circular intersection on the surface of the sphere, encloses an area equivalent to 1% of the area on ONE hemisphere. (1% is the default, and commonly used value, however, the user can set different values if desired, between 0.5% and 5%, using the Count Circle option in the Stereonet Options dialog.

This circle (or cone) is then centered on each grid point on the sphere, and the pole vectors falling within the cone are counted (using either the Schmidt or Fisher influence for each pole), and the total at each grid point is divided by the total population, to obtain a concentration value (density) at each grid point on the sphere.

Once the grid point densities on the sphere have been calculated, the grid values are assigned to the associated grid points on the 2D stereonet. This square array of values is then contoured using linear interpolation in 2 dimensions, to generate the continuous contours that you see in Dips.

Notes:

The "floating cone" used in Dips is analagous, in 3D, to the "floating circle" counting method described in Rock Slope Engineering. However, distortion and data loss are completely eliminated, since the counting is done on the sphere and not on the projection. (2D grid cell methods such as the Denness method, are NOT used in Dips.)

If you are using Traverses in your Dips file, note that both WEIGHTED (Terzaghi bias correction applied) and UNWEIGHTED densities are calculated at each grid point. This produces the WEIGHTED contour plot in Dips. If you are not using Traverses, then weighted and unweighted plots will appear the same.

The formulae for the confidence and variability limits used in Dips, are as follows:

Variability limit angle: cos(alpha) = 1.0 + log(1-P)/(K)

Confidence limit angle: cos(alpha) = 1.0 + log(1-P)/(RK)

where P is one of 2 probabilities:

For variability, it is the probability that a vector selected at random makes a solid angle of theta with the calculated mean.

For confidence, it is the probability that the calculated mean is within theta of the true population mean.

P in the above formulae ranges from 0 to 1 (i.e. 0 % to 100%)

K and R are defined as follows:

K is Fisher's constant = (N - 1) / (N - R)

where N is the total length of pole vectors in a set, and R is the length of the resultant vector (upon vector addition of all poles in the set)

NOTE:

The numerical value of Fisher's constant K, for each set that you have created, can be found in the Info Viewer listing in Dips.

The values of N and R are derived from the mean vector calculation for a set. Since we consider a sphere of unit radius, each pole vector is of unit length, and therefore the value of N is numerically equal to the number of poles in the set. The value of R is the length of the resultant vector, upon vector addition of all poles in a set (therefore the value of R is always less than or equal to the value of N). The orientation of vector R, is the mean orientation for the set.